Solve (x+1)(2+x)=156

$$(x+1)(2+x)=156$$

Solve for x

$$x=-14$$

$$x=11$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+1 by 2+x and combine like terms.

$$3x+x^{2}+2=156$$

Subtract 156 from both sides.

$$3x+x^{2}+2-156=0$$

Subtract 156 from 2 to get -154.

$$3x+x^{2}-154=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+3x-154=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and -154 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-154\right)}}{2}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-154\right)}}{2}$$

Multiply -4 times -154.

$$x=\frac{-3±\sqrt{9+616}}{2}$$

Add 9 to 616.

$$x=\frac{-3±\sqrt{625}}{2}$$

Take the square root of 625.

$$x=\frac{-3±25}{2}$$

Now solve the equation x=\frac{-3±25}{2} when ± is plus. Add -3 to 25.

$$x=\frac{22}{2}$$

Divide 22 by 2.

$$x=11$$

Now solve the equation x=\frac{-3±25}{2} when ± is minus. Subtract 25 from -3.

$$x=-\frac{28}{2}$$

Divide -28 by 2.

$$x=-14$$

The equation is now solved.

$$x=11 x=-14$$

Steps for Completing the Square

Use the distributive property to multiply x+1 by 2+x and combine like terms.

$$3x+x^{2}+2=156$$

Subtract 2 from both sides.

$$3x+x^{2}=156-2$$

Subtract 2 from 156 to get 154.

$$3x+x^{2}=154$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+3x=154$$

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=154+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=154+\frac{9}{4}$$

Add 154 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{625}{4}$$

Factor x^{2}+3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{625}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{625}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{25}{2} x+\frac{3}{2}=-\frac{25}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=11 x=-14$$